The hyperbolicity conjecture for semi-symmetric polynomials

For $n,p\in\mathbb{N}$, define the $p$-homogeneous semi-symmetric polynomial

$$s_{n,p}(x,\lambda)=\sum_{k=0}^p\frac{\sigma_k(x)\sigma_{p-k}(\lambda)}{\binom{n}{k}},$$

for $x\in\mathbb{R}^n$ and $\lambda\in\mathbb{R}^p$. Let $\mathbb{I}$ denote the all-ones direction, and call a homogeneous polynomial $\mathbb{I}$-hyperbolic when its restrictions along lines in that direction have only real roots. Semi-symmetric hyperbolicity conjecture. The polynomial $s_{n,p}$ is $\mathbb{I}$-hyperbolic. The paper proves that this is equivalent to the corresponding assertion for $s_{p,p}$ and explains that it implies the real-rootedness concavity conjecture; the assertion itself is left open.